Any half decent containerized setup should support zero-downtime deploys. Usually it involves bringing up new containers and signaling the existing containers to begin draining connections.

For most workloads it should be entirely possible to deploy a new stateless config and not need to resort to using mutable state for critical infrastructure.

If you have long-lived, stateful connections (perhaps for live streams) then I can see why re-configuring in place would be desirable, but in my experience that's pretty rare.